Wing Gonzalo Camacho's 76th-minute try, converted by Nick Evans, inched Quins home at Cardiff City Stadium - winning European rugby's second-tier knockout tournament for a record third time.
Stade looked to have done enough through four Lionel Beauxis penalties and drop-goals from centre Mathieu Bastareaud and full-back Martin Rodriguez.
But Quins conjured a special finish as prop Joe Marler and scrum-half Danny Care completed the hard yards for Camacho to cross and hand Evans his match-winning chance.
Evans earlier kicked four penalties for a 14-point haul, and Stade now need Northampton to win tomorrow's Heineken final against Irish heavyweights Leinster in the Welsh capital.
If Saints topple the favourites and become England's first European champions since 2007, then Stade will join Quins in Heineken action next term.
But a Leinster triumph would leave Stade floundering and send Irish province Connacht through instead.
Quins, though, have no such worries after lifting the Amlin Challenge Cup for the third time following wins in 2001 and 2004.
And it guarantees England will have seven teams in the 2011-12 Heineken Cup after France failed to produce a finalist for this season's blue riband European tournament.
Stade were mentally shattered at the end, particularly their England international forwards Tom Palmer and James Haskell, who both produced top-class displays.
But their England colleague Care made the telling contribution of a stop-start final, breaking deep into Stade territory and then having the presence of mind to send through a teasing kick that Camacho latched on to.
Evans, despite a testing angle, showed his class to convert with nerveless aplomb, and Stade simply ran out of options during the frantic closing stages.
Quins were unchanged from the side that stunned semi-final opponents Munster in Limerick three weeks ago, while Stade included lock Palmer and flanker Haskell, with ex-Newcastle wing Ollie Phillips among their replacements.
Stade spent most of the opening 10 minutes in defensive mode, and Quins deservedly went ahead through a long-range Evans penalty, although Beauxis cancelled that out when he bisected the posts from 50 metres.
A lively start kept referee George Clancy busy, with a handful of niggling off-the-ball skirmishes underlining ferocious commitment from teams not prepared to give an inch.
Evans' second penalty nudged Quins 6-3 ahead, and with clear try-scoring opportunities at a premium, the former New Zealand All Black proved a dominant figure throughout the first half.
Evans and Beauxis exchanged further penalties, but both sides were then left to rue missed chances - Quins after a sharp Care break, and Stade when skipper Sergio Parisse ignored an unmarked Palmer inside him just 10 metres from the Londoners' line.
It was a fast and furious contest, yet neither team could show the required composure in attack to reward impressive approach work.
But Stade probably trooped off marginally the happier, despite conceding a 9-6 interval scoreline.
And Quins found themselves under sustained pressure early in the second period as Stade went through several attacking phases, although their only reward was a third Beauxis penalty that tied things up.
Bastareaud's classy drop-goal then edged Stade ahead for the first time, underlining that Quins needed to tighten up as their opponents began to boss the forward exchanges.
Evans ensured a tense finish with his fourth penalty from five attempts as Stade appeared unable to play the territory game, while also continuing to infringe on a regular basis.
Such indiscipline allowed Quins to retain a foothold, but superb lineout work by Palmer set up a Stade counter-attack and spelled trouble for Quins.
Rodriguez's drop-goal took Stade 18-12 clear, and Quins' hopes looked to have finally disappeared, but then came Camacho's try at the final knockings, converted by Evans, and Stade were sunk.
For the French club, it was their third failure in a European final following Heineken losses in 2001 and 2005. Quins though, could begin a night of prolonged celebrations.
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